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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13982–13988
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Discrete Cylindrical Vector Beam Generation from an Array of Optical Fibers

R. Steven Kurti, Klaus Halterman, Ramesh K. Shori, and Michael J. Wardlaw  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13982-13988 (2009)
http://dx.doi.org/10.1364/OE.17.013982


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Abstract

A novel method is presented for the beam shaping of far field intensity distributions of coherently combined fiber arrays. The fibers are arranged uniformly on the perimeter of a circle, and the linearly polarized beams of equal shape are superimposed such that the far field pattern represents an effective radially polarized vector beam, or discrete cylindrical vector (DCV) beam. The DCV beam is produced by three or more beams that each individually have a varying polarization vector. The beams are appropriately distributed in the near field such that the far field intensity distribution has a central null. This result is in contrast to the situation of parallel linearly polarized beams, where the intensity peaks on axis.

© 2009 Optical Society of America

1. Introduction

The propagation of electromagnetic fields from multiple fibers can result in complex far-field intensity profiles that depend intricately on the individual near field polarization and phase. Coherently phased arrays have been used in defense and communications systems for many years, where the antenna ensemble forms a diffraction pattern that can be altered by changing the antenna spacing, amplitude, and relative phase relationships. In the optical regime, similar systems have recently been formed by actively or passively locking the phases of two or more identical optical beams [1–4

1. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006). [CrossRef] [PubMed]

]. While coherently combined fiber lasers are increasingly gaining acceptance as sources for high power applications, most approaches involve phasing a rectangular or hexagonal grid of fibers with collimated beams that are linearly polarized along the same axis [5

5. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. 13(3), 480–486 (2007). [CrossRef]

]. The resultant far field diffraction pattern therefore is peaked in the center. If the polarization is allowed to vary, as it does in radial vector beams, the diffraction pattern vanishes in the center [6

6. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322 (2000). [CrossRef]

], which can result in a myriad of device applications, discussed below.

Several types of polarization states have been investigated, including radial and azimuthal [6–18

6. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322 (2000). [CrossRef]

]. These beams are used in mitigating thermal effects in high power lasers [19–21

19. I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. 32(1), 47–49 (2007). [CrossRef]

], laser machining [22

22. V. G. Niziev and A. V. Nestorov, and J. Phys, “D,” Appl. Phys. (Berl.) 32, 1455 (1999).

, 23

23. M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17(10), 1532 (1978). [CrossRef] [PubMed]

], and particle acceleration interactions [24

24. Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007). [CrossRef]

, 25

25. S. M. Iftiquar and J. Opt , “A tunable doughnut laser beam for cold-atom experiments,” B: Quantum Semiclass. Opt. 5(1), 40–43 (2003). [CrossRef]

]. They can even be used to generate longitudinal electric fields when tightly focused [26

26. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

,27

27. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

], and although they are usually formed in free space laser cavities using a conical lens, or axicon, they can also be formed and guided in fibers [28–30

28. T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1–2), 1–5 (2002). [CrossRef]

]. Typical methods for creating radially polarized beams fall into various categories: one scenario involves an intracavity axicon or another polarization-selective element in a laser resonator to generate the laser mode. It has been reported [31

31. D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Appl. Opt. 42(18), 3550–3554 (2003). [CrossRef] [PubMed]

] that four Gaussian beams, rotated intra-cavity, can be effectively combined on the resonator exit mirror. Other approaches involve taking a single beam and rotating the polarization of portions of the beam to create an inhomogeneously polarized beam (typically radial or azimuthal). Such elements as a polarization-selective mirror [8

8. T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006). [CrossRef] [PubMed]

,11

11. A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

,22

22. V. G. Niziev and A. V. Nestorov, and J. Phys, “D,” Appl. Phys. (Berl.) 32, 1455 (1999).

], or an aperture used with strong-birefringence rods [7

7. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]

,19–21

19. I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. 32(1), 47–49 (2007). [CrossRef]

] are also widely used.

Fig. 1. Example of the multiple fiber setup. A specific case of N = 3 is shown, where each beam is represented by a circle of radius a and arranged uniformly on a circle of radius R. Points within the ith hole are located by d = s i, + r, with the vector r originating at the center of each hole.

In this paper we investigate the generation of an effective radially-polarized beam by coherently combining Gaussian beams, each one with an appropriately oriented linear polarization. This new technique of generating a cylindrical vector beam has the advantage of simplicity, direct control over mode generation, and eliminates power propagation along the center axis for high power Gaussian beam applications. In each particular configuration of the fibers, experimental findings are in good agreement with predictions based on standard Fraunhofer diffraction theory.

2. Method and Experimental Setup

Our starting point in determining the propagation of polarized electromagnetic beams is the vector Helmholtz equation, ∇ × ∇ × E(ρ,z) - k 2 0 E(ρ,z) = 0. We use cylindrical coordinates, so that ρ is the usual transverse coordinate, k 0 = ω/c, and the usual sinusoidal time dependence has been factored out. Applying Lagrange’s formula permits the wave equation to be written in terms of the vector Laplacian: ∇2 E(ρ,z) + k 2 0 E(ρ,z) = 0, which can then be reduced to the paraxial wave equation. The paraxial solutions are then inserted into the familiar Fraunhofer diffraction integral, which for propagation at sufficiently large z, yields the electric field,

E(kx,ky)=exp(ik0ρ22z)exp(ik0z)iλzj=1N0ardr02πeik·(r+sj)E0(r+sj,0),
(1)

where E 0(r+s j,0) is the incident electric field in the plane of the fiber array. It is clear that longitudinal polarization is not considered here, consistent with the paraxial approximation. The far field intensity will be a complex diffraction pattern that depends on the individual beam intensity profile in the near field as well its polarization state and optical phase. A diagram illustrating the geometry used is shown in Fig. 1, where an array of three circular holes of radius a are equally distributed on the circumference of a circle of radius R. The center of each hole is located at s j = R r̂j, where r̂j = (x̂ cos θj + ŷ sin θj) is the unit radial vector, and r is the coordinate relative to each center. The center of each fiber is separated by an angle θj = 2π(j - 1)/N, where N is the number of beams. One can scale to any number of beams, constrained mainly by the radius R. For identical Gaussian beams, the incident field is expressed as, E 0 = E 0(r)r̂j, where E 0(r) = E 0 e -r2/w20. This permits Eq. (1) to be separated,

E(kx,ky)=(ρ,z)j=1Neik·sjr̂j,
(2)

where ℱ = E 0 k 0/za 0 rdr J 0(k 0 /z)e-r2/w20, and the prefactors that do not contribute to the intensity have been suppressed. It is clear from Eq. (2) that the far field transform preserves the radially symmetric polarization state of the system, as expected. Note that in the limit w 0/a ≫ 1, and using the relation xJ 0(x) = [x J 1(x)]′, we have ℱ(ρ,z) ≈ E 0(a/ρ)J 1(k 0 /z), which gives the expected Fraunhofer diffraction pattern for a circular aperture of radius a. In the opposite limit, where the Gaussian profile is narrow enough so that the aperture geometry has little effect, we have ℱ(ρ,z) ≈ E 0 k 0 w 2 0(2z) exp[-(k 0 ρ w 0/(2z))2]. For the multiple beam arrangements investigated, corresponding to N = 3,4,6, we then can calculate the intensity, IN (ρ,ϕ), explicitly:

3(ϕ)=3cos(3kyR)2cos(32kyR)cos(32kxR),
(3)
𝓚4(ϕ)=4[sin2(kxR)+sin2(kyR)],
(4)
K6(ϕ)=4[(cos(32kyR)sin(12kxR)+sin(kxR))2+3cos2(12kxR)sin2(32kyR)],
(5)

where IN = ℓ2(ρ,z)퓚N(ϕ;ρ,z), kx = (k 0/z)ρ cos ϕ and ky = (k 0/z)ρ sin ϕ. To rotate the array, one can perform a standard rotation r′ = 𝫡(ϕ′)r, so e.g., a π/4 rotation would give, 𝓚4(ϕ′) = 4(1 - cos(√2kxR) cos(√2kyR)).

We employ two different methods to create the DCV beams previously described. In each method, the experimental configuration utilizes collimated Gaussian beams that are in phase. The arrays are cylindrically symmetric both with respect to intensity and polarization. We focus the beams with a transform lens in order to simulate the far field intensity at the lens focus. The focal spot is then imaged onto a camera by a microscope objective in order to fill the CCD array. The laser source is a 100 mW continuous wave (CW) Nd:YAG operating at λ = 1.064μm.

The first method (for the case of N = 4) involves expanding a beam from the Nd:YAG by means of a telescope beam expander to a diameter of about 1cm. The diffractive optical element then creates an 8 × 8 array of beams. Four of the beams are reflected and made nearly parallel by a segmented mirror, and then linearly polarized upon passing through a set of four half-wave plates. The phases of each beam are made identical by having them pass through articulated glass slides.

The second method (for the case of N = 3 and 6), which is simpler for larger arrays, is shown in Fig. 2. In this method the Nd:YAG is coupled into a polarization maintaining fiber. This signal is then split into multiple copies by means of a lithium niobate waveguide 8-way splitter and 8-channel electro-optic modulator. Each path can then have a separate phase modulation to ensure proper phasing of the beams. These signals are then propagated through fiber and coupled out and collimated by a 1 in. lens. The lens is slightly overfilled so the Gaussian outputs of the fibers are truncated at 1.1 × (the e -1 radius).

Fig. 2. Diagram of the experimental setup for DCV beam generation. The oscillator output is split and phase modulated (S/PM). The output of each fiber is collimated by means of a lens, L1, and then the polarization is rotated by a half-wave plate (HWP). The beam ensemble is then focused to a point which is imaged by a microscope objective (MO) onto a charge-coupled-device (CCD) array.
Fig. 3. Far field intensity profiles of effectively radially polarized beams. The left panels correspond to the experimental images and the right panels represent the theoretical results. We consider a 3 (top row), 4 (middle row), and 6 (bottom row) beam arrangement. For N = 3, we take w 0 = 32 μm and R = 5w 0. For N = 4, w 0 = 42 μm, and R = 2.7w 0. For N = 6, we have w 0 = 43 μm and R = 3.8w 0. Note that in general, a radially polarized beam undergoes a discontinuity at the origin, and thus the intensity must vanish there.

3. Discussion

Experimental results for the radial vector beam generation found good agreement with theoretical calculations, as illustrated in Fig. 3. Clearly, the net interference patterns and symmetry correlate well with the corresponding calculations. For N = 4 (middle row), the central null is surrounded by a checkerboard peak structure. This pattern clearly differs from the linear polarization by a rotation of π/4, based on a simple phase argument. The N = 6 case is shown in the bottom row, where the intensity vanishes at the origin, followed by the formation of bright hexagonal rings. The 3 beam configuration is shown in the top row, where again there is satisfactory agreement between the calculated and measured results. Any observed discrepancies may be reduced with an appropriately incorporated feedback system.

A substantial fraction of the energy in the plots of Fig. 3 is contained within the first regions of high intensity peaks. To define an effective measure of this, we integrate the intensity over a circular region of radius ρ, and normalize it to the intensity integrated over the entire image plane. This fraction, 𝒰N, can be determined for a given number of fibers, N. If one considers a circular region that extends up to the primary peaked intensity patterns, it is often possible to accurately calculate this analytically. This is shown in particular for the N = 4 case, 𝒰4, where,

𝓤4k0ρw0248z2(e−2R2/w02−1){k0ρ[24+12w02R2J2(2Rk0ρ/z)+w02k0ρz2[k03ρ3w02z26k0ρ
+3J3(2Rk0ρ/z)(k02ρ2w02Rz2zw02R34zR)]]24zRJ1(2Rk0ρ/z)}.
(6)

Inserting the corresponding parameters, w 0 = 42μm and R = 2.7w 0 into Eq. (6), we find 𝒰4 ≈ 0.57, which effectively characterizes the dominant field distribution contained within the given circle. Performing an analogous calculation at the same radius for N = 6,8 and 10 yields 𝒰6 ≈ 0.69, 𝒰8 ≈ 0.81, and 𝒰10 ≈ 0.84, respectively. Thus, as expected, increasing the number of combined beams generally enhances the fraction, 𝒰N, and ultimately the desired radially polarized DCV beam emerges.

Although the polarization symmetry is expected to be preserved in the far field, the observed central null in the far field intensity distributions indirectly confirms that the beams have a radial polarization distribution. Our experimental technique required each beam’s near field polarization to be individually aligned and each beam was directly verified to have a linear polarization aligned radially from the center of the ensemble. Further direct evidence of the radial polarization could be obtained after passing the beams through a linear polarizer in different orientations, or alternatively through a λ/2 retardation plate in different orientations in conjunction with a polarizing beam splitter.

As for scaling, there is no fundamental limitation on the number of beams used. Several methods are available to incorporate a greater number of beams, including concentric beam placement (following the same beam-combining prescription described in this paper) that could scale in roughly the same pattern as a Bessel beam. It is also possible to use fiber lasers with active phasing, thus potentially scaling to many hundreds of beams [5

5. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. 13(3), 480–486 (2007). [CrossRef]

]. Our system also has a practical advantage over typical high power Gaussian beam applications which have the drawback of having the beam concentrated near the center, where the reflecting beam director (such as a telescope obscuration) resides. Our device on the other hand, has no power propagating along the center axis of the beam, and thus for high power applications requiring a center obscuration telescope beam director, our proposed system offers new advances.

Acknowledgments

References and links

1.

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006). [CrossRef] [PubMed]

2.

H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339 (2005). [CrossRef] [PubMed]

3.

T. B. Simpson, F. Doft, P. R. Peterson, and A. Gavrielides, “Coherent combining of spectrally broadened fiber lasers,” Opt. Express 15(18), 11731–11740 (2007). [CrossRef] [PubMed]

4.

A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]

5.

T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quant. El. 13(3), 480–486 (2007). [CrossRef]

6.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322 (2000). [CrossRef]

7.

G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]

8.

T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006). [CrossRef] [PubMed]

9.

N. Passilly, R. de Saint Denis, K. At-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984 (2005). [CrossRef]

10.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234 (1990). [CrossRef] [PubMed]

11.

A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]

12.

T. Hirayama, Y. Kozawa, T. Nakamura, and S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14(26), 12839–12845 (2006). [CrossRef] [PubMed]

13.

Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002). [CrossRef]

14.

K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

15.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1–3), 12–21 (2005). [CrossRef]

16.

J. J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 from a wavelength-tunable laser,” IEEE J. Quantum Electron. 10(2), 125–127 (1974). [CrossRef]

17.

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30(22), 3063–3065 (2005). [CrossRef] [PubMed]

18.

Y. I. Salamin, “Fields of a radially polarized Gaussian laser beam beyond the paraxial approximation,” Opt. Lett. 31(17), 2619–2621 (2006). [CrossRef] [PubMed]

19.

I. Moshe, S. Jackel, A. Meir, Y. Lumer, and E. Leibush, “2 kW, M2 ¡ 10 radially polarized beams from aberration-compensated rod-based Nd:YAG lasers,” Opt. Lett. 32(1), 47–49 (2007). [CrossRef]

20.

I. Moshe, S. Jackel, and A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28(10), 807–809 (2003). [CrossRef] [PubMed]

21.

M. Roth, E. Wyss, H. Glur, and H. P. Weber, “Generation of radially polarized beams in a Nd:YAG laser with self-adaptive overcompensation of the thermal lens,” Opt. Lett. 30(13), 1665 (2005). [CrossRef] [PubMed]

22.

V. G. Niziev and A. V. Nestorov, and J. Phys, “D,” Appl. Phys. (Berl.) 32, 1455 (1999).

23.

M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17(10), 1532 (1978). [CrossRef] [PubMed]

24.

Y. I. Salamin, “Mono-energetic GeV electrons from ionization in a radially polarized laser beam,” Opt. Lett. 32(1), 90–92 (2007). [CrossRef]

25.

S. M. Iftiquar and J. Opt , “A tunable doughnut laser beam for cold-atom experiments,” B: Quantum Semiclass. Opt. 5(1), 40–43 (2003). [CrossRef]

26.

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

27.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

28.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun. 203(1–2), 1–5 (2002). [CrossRef]

29.

G. Volpe and D. Petrov, “Generation of cylindrical vector beams with few-mode fibers excited by Laguerre Gaussian beams,” Opt. Commun. 237(1–3), 89–95 (2004). [CrossRef]

30.

J. L. Li, K. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, “Generation of radially polarized mode in Yb fiber laser by using a dual conical prism,” Opt. Lett. 31(20), 2969–2971 (2006). [CrossRef] [PubMed]

31.

D. J. Armstrong, M. C. Phillips, and A. V. Smith, “Generation of radially polarized beams with an image-rotating resonator,” Appl. Opt. 42(18), 3550–3554 (2003). [CrossRef] [PubMed]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(230.5440) Optical devices : Polarization-selective devices
(060.3510) Fiber optics and optical communications : Lasers, fiber

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: June 8, 2009
Revised Manuscript: July 14, 2009
Manuscript Accepted: July 17, 2009
Published: July 28, 2009

Citation
R. S. Kurti, Klaus Halterman, Ramesh K. Shori, and Michael J. Wardlaw, "Discrete cylindrical vector beam generation from an array of optical fibers," Opt. Express 17, 13982-13988 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13982


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References

  1. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14(25), 12188–12195 (2006). [CrossRef] [PubMed]
  2. H. Bruesselbach, D. C. Jones, M. S. Mangir, M. I. Minden, and J. L. Rogers, “Self-organized coherence in fiber laser arrays,” Opt. Lett. 30(11), 1339 (2005). [CrossRef] [PubMed]
  3. T. B. Simpson, F. Doft, P. R. Peterson, and A. Gavrielides, “Coherent combining of spectrally broadened fiber lasers,” Opt. Express 15(18), 11731–11740 (2007). [CrossRef] [PubMed]
  4. A. Shirakawa, T. Saitou, T. Sekiguchi, and K. Ueda, “Coherent addition of fiber lasers by use of a fiber coupler,” Opt. Express 10(21), 1167–1172 (2002). [PubMed]
  5. T. M. Shay, V. Benham, J. T. Baker, A. D. Sanchez, D. Pilkington, and C. A. Lu, “Self-Synchronous and Self-Referenced Coherent Beam Combination for Large Optical Arrays,” IEEE J. Sel. Top. Quantum Electron. 13(3), 480–486 (2007). [CrossRef]
  6. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77(21), 3322 (2000). [CrossRef]
  7. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]
  8. T. Moser, J. Balmer, D. Delbeke, P. Muys, S. Verstuyft, and R. Baets, “Intracavity generation of radially polarized CO2 laser beams based on a simple binary dielectric diffraction grating,” Appl. Opt. 45(33), 8517–8522 (2006). [CrossRef] [PubMed]
  9. N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J.-F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984 (2005). [CrossRef]
  10. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234 (1990). [CrossRef] [PubMed]
  11. A. V. Nesterov and V. G. Niziev, J. Phys. D Appl. Phys. 33(15), 1817–1822 (2000). [CrossRef]
  12. T. Hirayama, Y. Kozawa, T. Nakamura, and S. Sato, “Generation of a cylindrically symmetric, polarized laser beam with narrow linewidth and fine tunability,” Opt. Express 14(26), 12839–12845 (2006). [CrossRef] [PubMed]
  13. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27(5), 285–287 (2002). [CrossRef]
  14. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]
  15. T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252(1-3), 12–21 (2005). [CrossRef]
  16. J. J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 from a wavelength-tunable laser,” IEEE J. Quantum Electron. 10(2), 125–127 (1974). [CrossRef]
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